Trig word problems

Violagirl

Junior Member
Joined
Mar 9, 2008
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87
Hi I've been having trouble with these 3 word problems and would really appreciate some help on getting started or getting to the next step. Thanks!! :)

1. A spin balancer rotates the wheel if a car at 480 revolutions per minute. If one diameter of the wheel is 26 inches, what road speed is being tested? Express your answer in miles per hour. At how many revolutions per minute should the balancer be set to test a road speed of 80 miles per hour?

For this one, I know that some conversions are involved such as 1 hour=60 min to cancel it out. And also, for the second part, you can divide 480 by 80 to get 6 miles per hour. Otherwise I'm not sure how to continue on. All the others I need help getting started on. I'm new to Trigonometry and am confused on these problems.

2. Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. Express the length L of the line segment to show it as a function of the angle theta.

3. The Freedom Tower is to be the centerpiece of the rebuilding of the World Trade Center in New York City. The tower will be 1776 feet tall (not including a broadcast antenna). The angle of elevation from the base of an office buidling nearby to the top of the tower is 34 degrees. The angle of evelation from the helipad on the roof of the office building to the top of the tower is 20 degrees.

A) How far away is the office building from the Freedom Tower? Assume the side of the tower is vertical. Round to the nearest foot.

B) How tall is the office building? Round to the nearest foot.
 
Hello, Violagirl!

3. The Freedom Tower is to be the centerpiece of the rebuilding of the WTC in New York.
The tower will be 1776 feet tall (not including a broadcast antenna). .The angle
of elevation from the base of a nearby office buidling to the top of the tower is 34 degrees.
The angle of evelation from the roof of the office building to the top of the tower is 20 degrees.

A) How far away is the office building from the Freedom Tower? Round to the nearest foot.

B) How tall is the office building? Round to the nearest foot.
Code:
                      * C
                  * * |
              *   *   | 1776-y
          *20d  *     |
    A * - - - * - - - + D
      |     *         |
    y |   *           | y
      | *34d          |
    B * - - - - - - - * E
              x

\(\displaystyle AB\text{ is the office building: }\,y \,=\,AB \,=\,DE\)

\(\displaystyle CE\text{ is the Freedom Tower: }\,CE = 1776,\;CD \,=\,1776-y\)

\(\displaystyle \angle CBE \,=\,34^o,\;\angle CAD \,=\,20^o\)

\(\displaystyle \text{Let }x \,=\,BE \,=\,AD\)


\(\displaystyle \text{In right triangle }CEB\!:\;\;\tan34^o \:=\:\frac{1776}{x} \quad\Rightarrow\quad x \:=\:\frac{1776}{\tan34^o} \:\approx\:2663\text{ ft}\quad (A)\)


\(\displaystyle \text{In right triangle }CDA\!:\;\;\tan20^o \:=\:\frac{1776-y}{2663}\)

. . \(\displaystyle \text{Then: }\;y \:=\:1776 - 2663\tan20^o \;\approx\;807\text{ ft}\quad (B)\)

 
Hello again, Violagirl!

I'm sure there was a diagram for #2.
Lucky for both of us, I'm familiar with this problem.


2. Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle.
Express the length L of the line segment to show it as a function of the angle theta.
Code:
                              C
      *-----------------------*------
      |                    *  :
      |                 *     : 3
      |          B   * @      :
      |           *-----------+------
      |        *  |          E
      |     *     |
      |  * @      |
    A * - - - - - + D
      |     4     |

\(\displaystyle \text{I assume that }L\text{ is the line segment }AC.\)


\(\displaystyle \text{In right triangle }BDA\!:\;\;\cos\theta \,=\,\frac{4}{AB} \quad\Rightarrow\quad AB \,=\,\frac{4}{\cos\theta}\)

\(\displaystyle \text{In right triangle }CEB\!:\;\;\sin\theta \,=\,\frac{3}{BC} \quad\Rightarrow\quad BC \,=\,\frac{3}{\sin\theta}\)


\(\displaystyle \text{Therefore: }\;L \;=\;AB + BC \;=\;\frac{4}{cos\theta} + \frac{3}{\sin\theta} \;=\;4\sec\theta + 3\csc\theta\)

 
Violagirl said:
Hi I've been having trouble with these 3 word problems and would really appreciate some help on getting started or getting to the next step. Thanks!! :)

1. A spin balancer rotates the wheel if a car at 480 revolutions per minute.
a) If one diameter of the wheel is 26 inches, what road speed is being tested? Express your answer in miles per hour.
b)At how many revolutions per minute should the balancer be set to test a road speed of 80 miles per hour?

For this one, I know that some conversions are involved such as 1 hour=60 min to cancel it out.
And also, for the second part, you can divide 480 by 80 to get 6 miles per hour. ---- Incorrect

do you know the following equation from Physics/Dynamics:

v = w * r

v = linear speed (in/sec)

w = angular speed (radians/sec)

r = radius (in)

In your case (part a)

w = 480/60 * 2? radians/sec

r = 13 in

v = 13 * 8 * 2? in/sec = 13 * 8 * 2? * 3600 in/hr
now continue....


Otherwise I'm not sure how to continue on. All the others I need help getting started on. I'm new to Trigonometry and am confused on these problems.

2. Two hallways, one of width 3 feet, the other of width 4 feet, meet at a right angle. Express the length L of the line segment to show it as a function of the angle theta.

3. The Freedom Tower is to be the centerpiece of the rebuilding of the World Trade Center in New York City. The tower will be 1776 feet tall (not including a broadcast antenna). The angle of elevation from the base of an office buidling nearby to the top of the tower is 34 degrees. The angle of evelation from the helipad on the roof of the office building to the top of the tower is 20 degrees.

A) How far away is the office building from the Freedom Tower? Assume the side of the tower is vertical. Round to the nearest foot.

B) How tall is the office building? Round to the nearest foot.
 
Hello, Violagirl!

1. A spin balancer rotates the wheel of a car at 480 revolutions per minute.
(a) If the diameter of the wheel is 26 inches, what road speed is being tested?
Express your answer in miles per hour.

\(\displaystyle \text{"480 rev/min" can be written: }\:\frac{480\text{ rev}}{1\text{ min}}\)


\(\displaystyle \text{The circumference of the wheel is: }\:\pi d \:=\:\pi(26) \:\approx\:81.68\text{ inches.}\)

\(\displaystyle \text{In one revolution, a point on the wheel moves 81.68 inches.}\)

\(\displaystyle \text{In one minute, the point moves: }\:\frac{480\:\rlap{///}\text{rev}}{1\text{ min}} \times \frac{81.68\text{ in}}{1\:\rlap{///}\text{rev}} \;=\;\frac{39,\!206.4\text{ in}}{1\text{ min}}\)

. . \(\displaystyle \text{This is: }\:\frac{39,\!206.4\:\rlap{//}\text{in}}{1\text{ min.}} \times \frac{1\text{ ft}}{12\:\rlap{//}\text{in}} \;=\;\frac{3,\!267.2\text{ ft}}{1\text{ min}}\)

. . \(\displaystyle \text{which equals: }\;\frac{3,\!267.2\text{ ft}}{1\:\rlap{///}\text{min}} \times \frac{60\:\rlap{///}\text{min}}{1\text{ hour}} \:=\:\frac{196,\!032\text{ ft}}{1\text{ hour}}\)

. . \(\displaystyle \text{which equals: }\;\frac{196,\!032\:\rlap{/}\text{ft}}{1\text{ hour}} \times \frac{1\text{ mile}}{5280\:\rlap{/}\text{ft}} \;=\;\frac{37.12727\text{ miles}}{1\text{ hour}} \;\approx\;37.1\text{ mph}\)




(b) At how many revolutions per minute should the balancer be set
to test a road speed of 80 miles per hour?

We simply "run the fractions backwards."

\(\displaystyle \frac{80\text{ miles}}{1\text{ hour}} \times \frac{5280\text{ ft}}{1\text{ mile}} \times \frac{12\text{ in}}{1\text{ ft}} \times \frac{1\text{ rev}}{81.68\text{ in}} \times \frac{1\text{ hour}}{60\text{ min}} \;\approx\;1034.3\text{ rev/min}\)

 
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